By integrating the Boltzmann equation over 
, we obtain the
continuity
equation
We can write the Fourier transform of the current
as a product 
. If 
is
analytic, we can write
which defines the time dependent transport coefficients
.
Upon substituting this back into the
continuity equation, we obtain a
generalization of the transport equation (
) having time
dependent transport coefficients:
Taking the Fourier transform of this equation, and dividing by
, the Fourier transform of n, one gets
The individual transport coefficients can be extracted from this power series by taking the lth derivative of this at the origin of k-space. Defining the operation
the transport coefficients can be expressed as
In Kumar et al. [1980], these coefficients are
identified with the time derivatives of certain correlation
functions. In particular, 
is the logarithmic
time derivative of the number of charged particles, and
is the velocity of the
centroid of the swarm.
